End-of-Unit Assessment

1.

Select all the true statements.

<p>Three rows of shapes. First row contains six smiley faces. Second row contains two triangles. Third row contains four squares. </p>

A.

The ratio of triangles to squares is 2 to 4.

B.

The ratio of squares to smiley faces is $6:4$ .

C.

The ratio of smiley faces to triangles is 6 to 4.

D.

There are two squares for every triangle.

E.

There are two triangles for every smiley face.

F.

There are three smiley faces for every triangle.

Answer: A, D, F

Teaching Notes

Students who select Statement B have the ratio backward ( $6:4$ vs. $4:6$ ). Students who fail to select any of the Statements A, D, or F may be confused about ratio language, either reversing the order of comparison or not understanding the general concept. Students who select Statement C might be mistakenly looking at the ratio of smiley faces to squares. Students who select Statement E are probably noticing that there are indeed two triangles, without understanding what “two triangles for every smiley face” means.

2.

Select all the ratios that are equivalent to $8:6$ .

A.

$4:3$

B.

$6:8$

C.

$16:12$

D.

$10:8$

E.

$7:5$

Answer: A, C

Teaching Notes

Students who select Statement B have the order of the ratio reversed. Students who select Statements D and E may be mistaking pairs of numbers with a common difference (in this case, a difference of 2) for equivalent ratios. Students who fail to select Statement A may think an equivalent ratio must use multiples of the numbers in the ratio. Students who fail to select Statement C may be unclear about the concept or may have made an arithmetic error.

3.

A mixture of purple paint contains 6 teaspoons of red paint and 15 teaspoons of blue paint. To make the same shade of purple paint using 35 teaspoons of blue paint, how much red paint would you need? Use the double number line diagram to help if needed.

Double number line. Red paint, teaspoons. Blue paint, teaspoons.

A.

12 teaspoons

B.

14 teaspoons

C.

18 teaspoons

D.

26 teaspoons

Answer:

14 teaspoons

Teaching Notes

Students who select Statement A have doubled the amount of red paint, possibly because the amount of blue paint has approximately doubled. Students who select Statement C may be thinking that the tick marks on the top number line are spaced 3 units apart, when in fact they must be spaced 2 units apart. Students who select Statement D may be using a constant difference between the blue paint and red paint, or (equivalently) using a scale of 1 tick mark equals 5 units for both number lines.

4.

Lin rode her bike 2 miles in 8 minutes. She rode at a constant speed. Complete the table to show the time it took her to travel different distances at this speed.

distance traveled (miles)	elapsed time (minutes)
2	8
1
	1

Answer:

distance traveled (miles)	elapsed time (minutes)
2	8
1	4
$\frac14$	1

Teaching Notes

This problem asks students to find two unit rates using a table—minutes per mile and miles per minute.

5.

Three movie tickets cost $36. At this rate, what is the cost per ticket?
Three ice cream cones cost $8.25. At this rate, how much do 2 ice cream cones cost?
Three bananas cost $0.99. At this rate, how much do 5 bananas cost?

Answer:

$12
$5.50
$1.65

Teaching Notes

The numbers in this problem are messy enough that most students will need to calculate the unit price first, then multiply the unit price by the number of items. Monitor for responses that involve calculating the unit price but do not go further.

6.

A bag contains 120 marbles. Some are red and the rest are black. There are 19 red marbles for every black marble. How many red marbles are in the bag? Explain your reasoning.

Answer:

114 red marbles. Sample reasoning: Make a table in which the first row has 19 red marbles and 1 black marble. To find the row where the total number of marbles is 120, multiply the first row by 6, giving 114 red marbles and 6 black marbles, which add to the correct total.

Minimal Tier 1 response:

Work is complete and correct.
Sample: $19 \boldcdot 6 = 114$ , so there are 114 red marbles and 6 black marbles.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Red and black marbles are in the correct ratio but do not add up to 120. Correct answer is given but with no work or explanation shown. Work involves some correct reasoning about equivalent ratios with some errors.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not show evidence of an understanding of equivalent ratios. Answer is incorrect and no work is shown.

Teaching Notes

Students will need to use some guesswork to find a pair of numbers that are in the correct ratio and that add up to 120.

7.

To make an orange-flavored drink, Noah mixes 4 scoops of powder with 6 cups of water. Andre mixes 5 scoops of powder with 8 cups of water.

Create a double number line or a table that shows different amounts of powder and water that taste the same as Noah’s mixture.
Create a double number line or a table that shows different amounts of powder and water that taste the same as Andre’s mixture.
How do their two mixtures compare in taste? Explain your reasoning.

Answer:

Sample response:

Noah’s recipe

scoops of powder	cups of water
4	6
2	3
1	1.5

Andre’s recipe

scoops of powder	cups of water
5	8
10	16
1	1.6

Noah’s recipe tastes stronger. Sample reasoning: As shown in the tables, Noah’s recipe uses less water for the same amount of powder.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: some mixing up of the terms “cup” and “scoop.”
Sample:

See table for Noah’s recipe.
See table for Andre’s recipe.
Noah’s tastes stronger because for him, 1 cup of water needs $\frac23$ of a scoop of powder, but for Andre, 1 cup of water needs only $\frac58$ of a scoop of powder.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: A good explanation to Part C is based on incorrect powder-water combinations found in Parts A and B.
Sample errors: Work shows arithmetic errors in otherwise reasonable tables or double number lines. Explanation for Part C is on track but leaves the reader to connect too many dots, for instance, “Noah’s tastes stronger because $\frac23 >\frac58$ .”

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Acceptable errors: A good explanation to Part C is based on incorrect powder-water combinations found in Parts A and B.
Sample errors: Representations in Parts A and B do not show evidence of an understanding of equivalent ratios. Correct representations are used in Parts A and B but explanation in Part C is missing or does not involve a comparison of unit rates.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: None of the work contains evidence of an understanding of equivalent ratios.

Teaching Notes

Make sure that students include at least two different powder-water combinations in their representations for each type of drink.